Round off your calculations to four decimal places. Estimate ∫0^2 (d x)/(√(4+x^2)) by: (a) the t

Round off your calculations to four decimal places. Estimate...

Key Concepts

Rounding and Precision

Rounding and precision are important aspects in numerical calculations, especially when an answer must be reported to a specific number of decimal places. This process ensures consistency and clarity in the presentation of numerical results, as well as managing the limitations of computer arithmetic in representing real numbers.

Simpson's Rule

Simpson's rule is a technique for numerical integration that approximates the integral of a function by fitting parabolas to segments of the function's graph. By using quadratic polynomials to approximate the function over intervals, Simpson's rule generally provides a more accurate estimate than methods like the trapezoidal rule when the integrand is sufficiently smooth.

Numerical Integration

Numerical integration refers to the process of calculating the approximate value of a definite integral using computational methods rather than analytical or closed-form evaluation. It is particularly useful when dealing with integrals of functions that are difficult or impossible to integrate exactly, offering a way to approximate the area under a curve through discretization of the interval of integration.

Trapezoidal Rule

The trapezoidal rule is a numerical integration method that approximates the area under a curve by dividing the integration interval into smaller subintervals and approximating the area under the curve in each subinterval by a trapezoid. The sum of these trapezoidal areas provides an estimate of the integral, and its accuracy depends on the number of subintervals used.

Transcript

00:01 We are going to estimate the interval from 0 to 2 of 1 over square root of 4 plus x cube using first trapezoidal rule with 4 sub -intervals in par a, and then simpson's rule with 2 sub -intervals in part b.

00:19 We will round off the calculations to 4 decimal places.

00:25 So let's start by defining the function we are integrating 1 over square root of 4, plus x cube and we consider that function on the close interval from 0 to 2.

00:50 Then in part a we recognize the lower limit of integration is 2 is 0, sorry, the upper limit of integration b is 2, numbers of intervals is 4, so the step size h which is the common distance between any two consecutive notes is defined as the length of the interval of integration b minus a over the numbers of intervals n in this case is equal to 2 minus 0 over 4 that is 1 half so each in this case step size is 1 half and with that we can say the nodes are given by x i equal the left and point of the interval of integration is 0 plus i times the step size h and that is i half 4 i from 0 up to 4.

01:53 So trapezoid the rule with 4 sub -intervals, 4 is h -half times f at the first node x0 plus 2 f at x1 plus 2f at x2 plus 2 f at x2 at x3 plus f at x3 plus f at x3 plus f at x1.

02:15 4 then travis other rule with force of intervals 24 is one half which is the value of h divided by 2 times f at the first note we use this expression here for the successive values of 8 of i so the first one is 0 plus 2 f at one half plus 2f at 2 half that is 1 plus 2 f at 2 half that is 1 plus 2 f at 3 three halves plus f at two.

02:58 Then t4 is one -fourth times.

03:05 Now we use the formula of the function.

03:07 It's one over square root of four plus x cubed.

03:10 So here is one over square root of four plus zero cube is one over square root of four.

03:15 That is one -half.

03:17 Plus two over square root of four plus one -half...

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